Theory of Linear Ordinary Differential Equations Bernd Schr oder - - PowerPoint PPT Presentation

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Theory of Linear Ordinary Differential Equations

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Definition

A linear n-th order differential equation is of the form an(x)y(n)(x)+an−1(x)y(n−1)(x)+···+a1(x)y′(x)+a0(x)y(x) = g(x), with an not being the constant function 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Definition

A linear n-th order differential equation is of the form an(x)y(n)(x)+an−1(x)y(n−1)(x)+···+a1(x)y′(x)+a0(x)y(x) = g(x), with an not being the constant function 0.

◮ It is called homogeneous if and only if g = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Definition

A linear n-th order differential equation is of the form an(x)y(n)(x)+an−1(x)y(n−1)(x)+···+a1(x)y′(x)+a0(x)y(x) = g(x), with an not being the constant function 0.

◮ It is called homogeneous if and only if g = 0. ◮ It is called inhomogeneous if and only if g = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Definition

A linear n-th order differential equation is of the form an(x)y(n)(x)+an−1(x)y(n−1)(x)+···+a1(x)y′(x)+a0(x)y(x) = g(x), with an not being the constant function 0.

◮ It is called homogeneous if and only if g = 0. ◮ It is called inhomogeneous if and only if g = 0. ◮ If, in an inhomogeneous equation, we replace the right side

g with 0, we obtain the corresponding homogeneous equation.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Definition

A linear n-th order differential equation is of the form an(x)y(n)(x)+an−1(x)y(n−1)(x)+···+a1(x)y′(x)+a0(x)y(x) = g(x), with an not being the constant function 0.

◮ It is called homogeneous if and only if g = 0. ◮ It is called inhomogeneous if and only if g = 0. ◮ If, in an inhomogeneous equation, we replace the right side

g with 0, we obtain the corresponding homogeneous equation. Note that the coefficients are functions. The results in this presentation apply to constant coefficient equations as well as Cauchy-Euler equations or the equations that are being solved with series solutions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Existence and Uniqueness

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Existence and Uniqueness

Every initial value problem of the form an(x)y(n)(x)+···+a1(x)y′(x)+a0(x)y(x) = g(x), y(x0) = y0, y′(x0) = y1, . . . y(n−1)(x0) = yn−1, where an is not the constant function 0 and all ai(x) and g(x) have continuous first derivatives, has a unique solution.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Existence and Uniqueness

Every initial value problem of the form an(x)y(n)(x)+···+a1(x)y′(x)+a0(x)y(x) = g(x), y(x0) = y0, y′(x0) = y1, . . . y(n−1)(x0) = yn−1, where an is not the constant function 0 and all ai(x) and g(x) have continuous first derivatives, has a unique solution. So, in some ways, the solutions look like n-dimensional space. We are interested in using this analogy.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof of the Existence and Uniqueness Theorem

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof of the Existence and Uniqueness Theorem

, p.510

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Furthering the Analogy Between Vectors and Solutions

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Furthering the Analogy Between Vectors and Solutions

Superposition Principle. Let y1 and y2 be solutions of the homogeneous linear differential equation an(x)y(n)(x)+an−1(x)y(n−1)(x)+···+a1(x)y′(x)+a0(x)y(x) = 0, and let c1 and c2 be real numbers. Then y(x) := c1y1(x)+c2y2(x) is a solution, too.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Furthering the Analogy Between Vectors and Solutions

Superposition Principle. Let y1 and y2 be solutions of the homogeneous linear differential equation an(x)y(n)(x)+an−1(x)y(n−1)(x)+···+a1(x)y′(x)+a0(x)y(x) = 0, and let c1 and c2 be real numbers. Then y(x) := c1y1(x)+c2y2(x) is a solution, too. So solutions of homogeneous equations have the same algebraic properties as vectors.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof of the Superposition Principle

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof of the Superposition Principle

an(x)y(n)

1 (x)+···+a1(x)y′ 1(x)+a0(x)y1(x) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof of the Superposition Principle

an(x)y(n)

1 (x)+···+a1(x)y′ 1(x)+a0(x)y1(x) = 0

an(x)y(n)

2 (x)+···+a1(x)y′ 2(x)+a0(x)y2(x) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof of the Superposition Principle

an(x)y(n)

1 (x)+···+a1(x)y′ 1(x)+a0(x)y1(x) = 0

an(x)y(n)

2 (x)+···+a1(x)y′ 2(x)+a0(x)y2(x) = 0

c1

• an(x)y(n)

1 (x)

+ ··· + a0(x)y1(x)

• =

c1 ·0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof of the Superposition Principle

an(x)y(n)

1 (x)+···+a1(x)y′ 1(x)+a0(x)y1(x) = 0

an(x)y(n)

2 (x)+···+a1(x)y′ 2(x)+a0(x)y2(x) = 0

c1

• an(x)y(n)

1 (x)

+ ··· + a0(x)y1(x)

• =

c1 ·0 +c2

• an(x)y(n)

2 (x)

+ ··· + a0(x)y2(x)

• =

c2 ·0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof of the Superposition Principle

an(x)y(n)

1 (x)+···+a1(x)y′ 1(x)+a0(x)y1(x) = 0

an(x)y(n)

2 (x)+···+a1(x)y′ 2(x)+a0(x)y2(x) = 0

c1

• an(x)y(n)

1 (x)

+ ··· + a0(x)y1(x)

• =

c1 ·0 +c2

• an(x)y(n)

2 (x)

+ ··· + a0(x)y2(x)

• =

c2 ·0 an(x)(c1y1)(n)(x) + ··· + a0(x)(c1y1)(x) =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof of the Superposition Principle

an(x)y(n)

1 (x)+···+a1(x)y′ 1(x)+a0(x)y1(x) = 0

an(x)y(n)

2 (x)+···+a1(x)y′ 2(x)+a0(x)y2(x) = 0

c1

• an(x)y(n)

1 (x)

+ ··· + a0(x)y1(x)

• =

c1 ·0 +c2

• an(x)y(n)

2 (x)

+ ··· + a0(x)y2(x)

• =

c2 ·0 an(x)(c1y1)(n)(x) + ··· + a0(x)(c1y1)(x) = +

• an(x)(c2y2)(n)(x)

+ ··· + a0(x)(c2y2)(x)

• =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof of the Superposition Principle

an(x)y(n)

1 (x)+···+a1(x)y′ 1(x)+a0(x)y1(x) = 0

an(x)y(n)

2 (x)+···+a1(x)y′ 2(x)+a0(x)y2(x) = 0

c1

• an(x)y(n)

1 (x)

+ ··· + a0(x)y1(x)

• =

c1 ·0 +c2

• an(x)y(n)

2 (x)

+ ··· + a0(x)y2(x)

• =

c2 ·0 an(x)(c1y1)(n)(x) + ··· + a0(x)(c1y1)(x) = +

• an(x)(c2y2)(n)(x)

+ ··· + a0(x)(c2y2)(x)

• =

an(x)(c1y1 +c2y2)(n)(x) + ··· + a0(x)(c1y1 +c2y2)(x) =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Handling Inhomogeneous Equations

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Handling Inhomogeneous Equations

For the linear inhomogeneous differential equation an(x)y(n)(x)+an−1(x)y(n−1)(x)+···+a1(x)y′(x)+a0(x)y(x) = g(x) let yh(x) denote the general solution of the corresponding homogeneous equation. Moreover let yp(x) be one particular solution of the inhomogeneous equation. Then the general solution of the inhomogeneous equation is y(x) = yp(x)+yh(x).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Handling Inhomogeneous Equations

For the linear inhomogeneous differential equation an(x)y(n)(x)+an−1(x)y(n−1)(x)+···+a1(x)y′(x)+a0(x)y(x) = g(x) let yh(x) denote the general solution of the corresponding homogeneous equation. Moreover let yp(x) be one particular solution of the inhomogeneous equation. Then the general solution of the inhomogeneous equation is y(x) = yp(x)+yh(x). So the theory of inhomogeneous equations is pretty much reduced to that of homogeneous equations.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof

an(x)y(n)

p (x)

+ ··· + a0(x)yp(x) = g(x)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof

an(x)y(n)

p (x)

+ ··· + a0(x)yp(x) = g(x) +

• an(x)y(n)

h (x)

+ ··· + a0(x)yh(x) =

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof

an(x)y(n)

p (x)

+ ··· + a0(x)yp(x) = g(x) +

• an(x)y(n)

h (x)

+ ··· + a0(x)yh(x) =

• an(x)(yp +yh)(n)(x)

+ ··· + a0(x)(yp +yh)(x) = g(x)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof

an(x)y(n)

p (x)

+ ··· + a0(x)yp(x) = g(x) +

• an(x)y(n)

h (x)

+ ··· + a0(x)yh(x) =

• an(x)(yp +yh)(n)(x)

+ ··· + a0(x)(yp +yh)(x) = g(x) and

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof

an(x)y(n)

p (x)

+ ··· + a0(x)yp(x) = g(x) +

• an(x)y(n)

h (x)

+ ··· + a0(x)yh(x) =

• an(x)(yp +yh)(n)(x)

+ ··· + a0(x)(yp +yh)(x) = g(x) and an(x)y(n)

i (x)

+ ··· + a0(x)yi(x) = g(x)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof

an(x)y(n)

p (x)

+ ··· + a0(x)yp(x) = g(x) +

• an(x)y(n)

h (x)

+ ··· + a0(x)yh(x) =

• an(x)(yp +yh)(n)(x)

+ ··· + a0(x)(yp +yh)(x) = g(x) and an(x)y(n)

i (x)

+ ··· + a0(x)yi(x) = g(x) −

• an(x)y(n)

p (x)

+ ··· + a0(x)yp(x) = g(x)

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Proof

an(x)y(n)

p (x)

+ ··· + a0(x)yp(x) = g(x) +

• an(x)y(n)

h (x)

+ ··· + a0(x)yh(x) =

• an(x)(yp +yh)(n)(x)

+ ··· + a0(x)(yp +yh)(x) = g(x) and an(x)y(n)

i (x)

+ ··· + a0(x)yi(x) = g(x) −

• an(x)y(n)

p (x)

+ ··· + a0(x)yp(x) = g(x)

• an(x)(yi −yp)(n)(x)

+ ··· + a0(x)(yi −yp)(x) =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Combinations of Vectors

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Combinations of Vectors

How do we actually know that several vectors “point in different directions”?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Combinations of Vectors

How do we actually know that several vectors “point in different directions”? Let v1, v2,..., vn be vectors. Then any sum

n

∑

i=1

ci

• vi = c1
• v1 +c2
• v2 +···+cn
• vn

with the ci being real numbers is called a linear combination

• f the vectors.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Independence for Vectors

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Independence for Vectors

A set of n vectors {

• v1,··· ,

vn} is called linearly dependent if and only if there are numbers c1,...,cn, which are not all zero, such that c1

• v1 +···+cn
• vn =

0, where 0 denotes the null vector, for which all components are zero.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Independence for Vectors

A set of n vectors {

• v1,··· ,

vn} is called linearly dependent if and only if there are numbers c1,...,cn, which are not all zero, such that c1

• v1 +···+cn
• vn =

0, where 0 denotes the null vector, for which all components are zero. If no such numbers exist, the set of vectors is called linearly

• independent. That is, a set of n vectors {
• v1,··· ,

vn} is called linearly independent if and only if the only numbers c1,··· ,cn, for which

n

∑

i=1

ci

• vi =

0 are c1 = c2 = ··· = cn = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1   1 1 3  +c2   2 4 2  +c3   3 −1 4   =    

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1   1 1 3  +c2   2 4 2  +c3   3 −1 4   =     1c1 + 2c2 + 3c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1   1 1 3  +c2   2 4 2  +c3   3 −1 4   =     1c1 + 2c2 + 3c3 = 1c1 + 4c2 − 1c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1   1 1 3  +c2   2 4 2  +c3   3 −1 4   =     1c1 + 2c2 + 3c3 = 1c1 + 4c2 − 1c3 = 3c1 + 2c2 + 4c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1 + 2c2 + 3c3 = c1 + 4c2 − c3 = 3c1 + 2c2 + 4c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1 + 2c2 + 3c3 = c1 + 4c2 − c3 = 3c1 + 2c2 + 4c3 = c1 + 2c2 + 3c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1 + 2c2 + 3c3 = c1 + 4c2 − c3 = 3c1 + 2c2 + 4c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1 + 2c2 + 3c3 = c1 + 4c2 − c3 = 3c1 + 2c2 + 4c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 4c2 − 5c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1 + 2c2 + 3c3 = c1 + 4c2 − c3 = 3c1 + 2c2 + 4c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 4c2 − 5c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1 + 2c2 + 3c3 = c1 + 4c2 − c3 = 3c1 + 2c2 + 4c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 4c2 − 5c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 13c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1 + 2c2 + 3c3 = c1 + 4c2 − c3 = 3c1 + 2c2 + 4c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 4c2 − 5c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 13c3 = 0 = c3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1 + 2c2 + 3c3 = c1 + 4c2 − c3 = 3c1 + 2c2 + 4c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 4c2 − 5c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 13c3 = 0 = c3 = c2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1 + 2c2 + 3c3 = c1 + 4c2 − c3 = 3c1 + 2c2 + 4c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 4c2 − 5c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 13c3 = 0 = c3 = c2 = c1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if the vectors (1,1,3), (2,4,2) and (3,−1,4) are linearly independent.

c1 + 2c2 + 3c3 = c1 + 4c2 − c3 = 3c1 + 2c2 + 4c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 4c2 − 5c3 = c1 + 2c2 + 3c3 = 2c2 − 4c3 = − 13c3 = 0 = c3 = c2 = c1, and the vectors are linearly independent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

2c1 − 1c2 + 3c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

2c1 − 1c2 + 3c3 = −2c1 + 2c2 − 2c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

2c1 − 1c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 = 2c1 − c2 + 3c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 = 2c1 − c2 + 3c3 = c2 + c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 = 2c1 − c2 + 3c3 = c2 + c3 = c2 + c3 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 = 2c1 − c2 + 3c3 = c2 + c3 = c2 + c3 = c2 = −c3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 = 2c1 − c2 + 3c3 = c2 + c3 = c2 + c3 = c2 = −c3, c1 = c2 −3c3 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 = 2c1 − c2 + 3c3 = c2 + c3 = c2 + c3 = c2 = −c3, c1 = c2 −3c3 2 , choose c3 = 1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 = 2c1 − c2 + 3c3 = c2 + c3 = c2 + c3 = c2 = −c3, c1 = c2 −3c3 2 , choose c3 = 1: c2 = −1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 = 2c1 − c2 + 3c3 = c2 + c3 = c2 + c3 = c2 = −c3, c1 = c2 −3c3 2 , choose c3 = 1: c2 = −1, c1 = −2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 = 2c1 − c2 + 3c3 = c2 + c3 = c2 + c3 = c2 = −c3, c1 = c2 −3c3 2 , choose c3 = 1: c2 = −1, c1 = −2. −2   2 −2 −4  −   −1 2 3  +   3 −2 −5   =    

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

2c1 − c2 + 3c3 = −2c1 + 2c2 − 2c3 = −4c1 + 3c2 − 5c3 = 2c1 − c2 + 3c3 = c2 + c3 = c2 + c3 = c2 = −c3, c1 = c2 −3c3 2 , choose c3 = 1: c2 = −1, c1 = −2. −2   2 −2 −4  −   −1 2 3  +   3 −2 −5   =     , and the vectors are linearly dependent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Why use Matrices?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Why use Matrices?

1 1 3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Why use Matrices?

1 1 3 2 4 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Why use Matrices?

1 1 3 2 3 4 −1 2 4

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Why use Matrices?

1 1 3 2 3 4 −1 2 4

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Matrices

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Matrices

Let m and n be positive integers. An m×n-matrix is a rectangular array of mn numbers aij, commonly indexed and written as follows. A=(ai,j) i = 1,...,m

j = 1,...,n

=          a11 a12 ··· a1(n−1) a1n a21 a22 ··· a2(n−1) a2n a31 a32 ··· a3(n−1) a3n . . . . . . a(m−1)1 a(m−1)2 ··· a(m−1)(n−1) a(m−1)n am1 am2 ··· am(n−1) amn          The index i is called the row index and the index j is called the column index.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determinants

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determinants

Let A = a11 a12 a21 a22

• be a 2×2 matrix. Then we define the

determinant of A to be det(A) := det a11 a12 a21 a22

• :=
• a11

a12 a21 a22

• := a11a22 −a12a21.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determinants

Let A = a11 a12 a21 a22

• be a 2×2 matrix. Then we define the

determinant of A to be det(A) := det a11 a12 a21 a22

• :=
• a11

a12 a21 a22

• := a11a22 −a12a21.

Let A = (aij)i,j=1,...,n be a square matrix and let Aij be the matrix

• btained by erasing the ith row and the jth column. Then the

determinant of A is defined recursively by det(A) := |A| :=

n

∑

j=1

(−1)i+jaij det(Aij) =

n

∑

i=1

(−1)i+jaij det(Aij), where the i in the first sum is an arbitrary row and the j in the second sum is an arbitrary column.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Uses of the Determinant

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Uses of the Determinant

• 1. The determinant gives the n-dimensional volume of the

parallelepiped spanned by the column vectors.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Uses of the Determinant

• 1. The determinant gives the n-dimensional volume of the

parallelepiped spanned by the column vectors.

• 2. The n-dimensional vectors

v1,..., vn are linearly independent if and only if det(

• v1,...,

vn) = 0, where (

• v1,...,

vn) denotes a matrix whose columns are the vectors vi.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Uses of the Determinant

• 1. The determinant gives the n-dimensional volume of the

parallelepiped spanned by the column vectors.

• 2. The n-dimensional vectors

v1,..., vn are linearly independent if and only if det(

• v1,...,

vn) = 0, where (

• v1,...,

vn) denotes a matrix whose columns are the vectors vi.

• 3. Computation of characteristic polynomials.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• +3·det

2 3 4 −1

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• +3·det

2 3 4 −1

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• +3·det

2 3 4 −1

• = 1·18

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• +3·det

2 3 4 −1

• = 1·18−1·2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• +3·det

2 3 4 −1

• = 1·18−1·2+3·(−14)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• +3·det

2 3 4 −1

• = 1·18−1·2+3·(−14)

= −26

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• +3·det

2 3 4 −1

• = 1·18−1·2+3·(−14)

= −26 = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   1 1 3  ,   2 4 2   and   3 −1 4   are linearly independent.

1 1 3 2 3 4 −1 2 4 det = 1·det 4 −1 2 4

• −1·det

2 3 2 4

• +3·det

2 3 4 −1

• = 1·18−1·2+3·(−14)

= −26 = 0 The vectors are linearly independent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

det   2 −1 3 −2 2 −2 −4 3 −5  

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

det   2 −1 3 −2 2 −2 −4 3 −5   = 2·det 2 −2 3 −5

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

det   2 −1 3 −2 2 −2 −4 3 −5   = 2·det 2 −2 3 −5

• −(−2)·det

−1 3 3 −5

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

det   2 −1 3 −2 2 −2 −4 3 −5   = 2·det 2 −2 3 −5

• −(−2)·det

−1 3 3 −5

• +(−4)·det

−1 3 2 −2

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

det   2 −1 3 −2 2 −2 −4 3 −5   = 2·det 2 −2 3 −5

• −(−2)·det

−1 3 3 −5

• +(−4)·det

−1 3 2 −2

• =

2·(−4)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

det   2 −1 3 −2 2 −2 −4 3 −5   = 2·det 2 −2 3 −5

• −(−2)·det

−1 3 3 −5

• +(−4)·det

−1 3 2 −2

• =

2·(−4)−(−2)(−4)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

det   2 −1 3 −2 2 −2 −4 3 −5   = 2·det 2 −2 3 −5

• −(−2)·det

−1 3 3 −5

• +(−4)·det

−1 3 2 −2

• =

2·(−4)−(−2)(−4)+(−4)(−4)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

det   2 −1 3 −2 2 −2 −4 3 −5   = 2·det 2 −2 3 −5

• −(−2)·det

−1 3 3 −5

• +(−4)·det

−1 3 2 −2

• =

2·(−4)−(−2)(−4)+(−4)(−4) =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if   2 −2 −4  ,   −1 2 3   and   3 −2 −5   are linearly independent.

det   2 −1 3 −2 2 −2 −4 3 −5   = 2·det 2 −2 3 −5

• −(−2)·det

−1 3 3 −5

• +(−4)·det

−1 3 2 −2

• =

2·(−4)−(−2)(−4)+(−4)(−4) = The vectors are linearly dependent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Combinations of Functions

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Combinations of Functions

We need to determine what it means that several functions “point in different directions”.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Combinations of Functions

We need to determine what it means that several functions “point in different directions”. Otherwise we would not be able to recognize that a family like yc1,c2(x) = c1 sin2(x)+c2

• 1−cos(2x)
• is not the general

solution of sin(x)y′′ −cos(x)y′ +2sin(x)y = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Combinations of Functions

We need to determine what it means that several functions “point in different directions”. Otherwise we would not be able to recognize that a family like yc1,c2(x) = c1 sin2(x)+c2

• 1−cos(2x)
• is not the general

solution of sin(x)y′′ −cos(x)y′ +2sin(x)y = 0. (The family has only one constant, because 2sin2(x) = 1−cos(2x).)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Combinations of Functions

We need to determine what it means that several functions “point in different directions”. Otherwise we would not be able to recognize that a family like yc1,c2(x) = c1 sin2(x)+c2

• 1−cos(2x)
• is not the general

solution of sin(x)y′′ −cos(x)y′ +2sin(x)y = 0. (The family has only one constant, because 2sin2(x) = 1−cos(2x).) Let f1,f2,...,fn be functions. Then any sum

n

∑

i=1

cifi = c1f1 +c2f2 +···+cnfn with the ci being real numbers is called a linear combination

• f the functions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Independence for Functions

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Independence for Functions

A set of n functions {f1,...,fn} is called linearly dependent if and only if there are numbers c1,...,cn, which are not all zero, such that c1f1 +···+cnfn = 0. That is, c1,...,cn must be such that for all x in the domain of f1,...,fn we have c1f1(x)+···+cnfn(x) = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Linear Independence for Functions

A set of n functions {f1,...,fn} is called linearly dependent if and only if there are numbers c1,...,cn, which are not all zero, such that c1f1 +···+cnfn = 0. That is, c1,...,cn must be such that for all x in the domain of f1,...,fn we have c1f1(x)+···+cnfn(x) = 0. If no such numbers exist, then the set of functions is called linearly independent. That is, a set of n functions {f1,...,fn} is called linearly independent if and only if the only numbers c1,...,cn, for which

n

∑

i=1

cifi = 0 are c1 = c2 = ··· = cn = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

The Wronskian

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

The Wronskian

Let f1,··· ,fn be (n−1) times differentiable functions. If the Wronskian W(f1,··· ,fn)(x) := det      f1(x) f2(x) ··· fn(x) f ′

1(x)

f ′

2(x)

··· f ′

n(x)

. . . . . . . . . f (n−1)

1

(x) f (n−1)

2

(x) ··· f (n−1)

n

(x)      is not equal to zero for some value of x, then {f1,··· ,fn} is a linearly independent set of functions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if t, et and tet are linearly independent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if t, et and tet are linearly independent.

det   t et tet 1 et tet +et et tet +2et  

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if t, et and tet are linearly independent.

det   t et tet 1 et tet +et et tet +2et   = t ·det et tet +et et tet +2et

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if t, et and tet are linearly independent.

det   t et tet 1 et tet +et et tet +2et   = t ·det et tet +et et tet +2et

• −1·det

et tet et tet +2et

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if t, et and tet are linearly independent.

det   t et tet 1 et tet +et et tet +2et   = t ·det et tet +et et tet +2et

• −1·det

et tet et tet +2et

• +0·det

et tet et tet +et

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if t, et and tet are linearly independent.

det   t et tet 1 et tet +et et tet +2et   = t ·det et tet +et et tet +2et

• −1·det

et tet et tet +2et

• +0·det

et tet et tet +et

• =

t

• te2t +2e2t −te2t −e2t

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if t, et and tet are linearly independent.

det   t et tet 1 et tet +et et tet +2et   = t ·det et tet +et et tet +2et

• −1·det

et tet et tet +2et

• +0·det

et tet et tet +et

• =

t

• te2t +2e2t −te2t −e2t

−

• te2t +2e2t −te2t

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if t, et and tet are linearly independent.

det   t et tet 1 et tet +et et tet +2et   = t ·det et tet +et et tet +2et

• −1·det

et tet et tet +2et

• +0·det

et tet et tet +et

• =

t

• te2t +2e2t −te2t −e2t

−

• te2t +2e2t −te2t

= te2t −2e2t

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if t, et and tet are linearly independent.

det   t et tet 1 et tet +et et tet +2et   = t ·det et tet +et et tet +2et

• −1·det

et tet et tet +2et

• +0·det

et tet et tet +et

• =

t

• te2t +2e2t −te2t −e2t

−

• te2t +2e2t −te2t

= te2t −2e2t = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Determine if t, et and tet are linearly independent.

det   t et tet 1 et tet +et et tet +2et   = t ·det et tet +et et tet +2et

• −1·det

et tet et tet +2et

• +0·det

et tet et tet +et

• =

t

• te2t +2e2t −te2t −e2t

−

• te2t +2e2t −te2t

= te2t −2e2t = 0 The functions are linearly independent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Defining the General Solution

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Defining the General Solution

The general solution of a differential equation is a family of functions so that for every initial value problem for the differential equation there is a unique choice of the coefficients that gives the solution of the initial value problem. A particular solution of a differential equation is one specific solution.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Defining the General Solution

The general solution of a differential equation is a family of functions so that for every initial value problem for the differential equation there is a unique choice of the coefficients that gives the solution of the initial value problem. A particular solution of a differential equation is one specific solution. In the theory, we typically work with initial value problems, because even this definition is a bit messy.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Solution Theorem for Linear Homogeneous Differential Equations

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations

logo1 Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem

Solution Theorem for Linear Homogeneous Differential Equations

The general solution of a linear homogeneous differential equation an(x)y(n)(x)+an−1(x)y(n−1)(x)+···+a1(x)y′(x)+a0(x)y(x) = 0, is of the form y(x) = c1y1(x)+···+cnyn(x), where {y1,··· ,yn} is a linearly independent set of particular solutions of the linear homogeneous differential equation.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations